# Properties

 Label 2028.4.b.d Level $2028$ Weight $4$ Character orbit 2028.b Analytic conductor $119.656$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,4,Mod(337,2028)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2028.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2028.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$119.655873492$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 156) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + \beta q^{5} - 16 \beta q^{7} + 9 q^{9} +O(q^{10})$$ q + 3 * q^3 + b * q^5 - 16*b * q^7 + 9 * q^9 $$q + 3 q^{3} + \beta q^{5} - 16 \beta q^{7} + 9 q^{9} - 34 \beta q^{11} + 3 \beta q^{15} + 14 q^{17} - 2 \beta q^{19} - 48 \beta q^{21} - 72 q^{23} + 121 q^{25} + 27 q^{27} + 102 q^{29} + 68 \beta q^{31} - 102 \beta q^{33} + 64 q^{35} - 193 \beta q^{37} - 125 \beta q^{41} + 140 q^{43} + 9 \beta q^{45} - 148 \beta q^{47} - 681 q^{49} + 42 q^{51} + 526 q^{53} + 136 q^{55} - 6 \beta q^{57} + 166 \beta q^{59} - 410 q^{61} - 144 \beta q^{63} - 298 \beta q^{67} - 216 q^{69} + 440 \beta q^{71} + 253 \beta q^{73} + 363 q^{75} - 2176 q^{77} - 640 q^{79} + 81 q^{81} - 690 \beta q^{83} + 14 \beta q^{85} + 306 q^{87} + 725 \beta q^{89} + 204 \beta q^{93} + 8 q^{95} + 223 \beta q^{97} - 306 \beta q^{99} +O(q^{100})$$ q + 3 * q^3 + b * q^5 - 16*b * q^7 + 9 * q^9 - 34*b * q^11 + 3*b * q^15 + 14 * q^17 - 2*b * q^19 - 48*b * q^21 - 72 * q^23 + 121 * q^25 + 27 * q^27 + 102 * q^29 + 68*b * q^31 - 102*b * q^33 + 64 * q^35 - 193*b * q^37 - 125*b * q^41 + 140 * q^43 + 9*b * q^45 - 148*b * q^47 - 681 * q^49 + 42 * q^51 + 526 * q^53 + 136 * q^55 - 6*b * q^57 + 166*b * q^59 - 410 * q^61 - 144*b * q^63 - 298*b * q^67 - 216 * q^69 + 440*b * q^71 + 253*b * q^73 + 363 * q^75 - 2176 * q^77 - 640 * q^79 + 81 * q^81 - 690*b * q^83 + 14*b * q^85 + 306 * q^87 + 725*b * q^89 + 204*b * q^93 + 8 * q^95 + 223*b * q^97 - 306*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 18 * q^9 $$2 q + 6 q^{3} + 18 q^{9} + 28 q^{17} - 144 q^{23} + 242 q^{25} + 54 q^{27} + 204 q^{29} + 128 q^{35} + 280 q^{43} - 1362 q^{49} + 84 q^{51} + 1052 q^{53} + 272 q^{55} - 820 q^{61} - 432 q^{69} + 726 q^{75} - 4352 q^{77} - 1280 q^{79} + 162 q^{81} + 612 q^{87} + 16 q^{95}+O(q^{100})$$ 2 * q + 6 * q^3 + 18 * q^9 + 28 * q^17 - 144 * q^23 + 242 * q^25 + 54 * q^27 + 204 * q^29 + 128 * q^35 + 280 * q^43 - 1362 * q^49 + 84 * q^51 + 1052 * q^53 + 272 * q^55 - 820 * q^61 - 432 * q^69 + 726 * q^75 - 4352 * q^77 - 1280 * q^79 + 162 * q^81 + 612 * q^87 + 16 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1015$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
0 3.00000 0 2.00000i 0 32.0000i 0 9.00000 0
337.2 0 3.00000 0 2.00000i 0 32.0000i 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.4.b.d 2
13.b even 2 1 inner 2028.4.b.d 2
13.d odd 4 1 156.4.a.b 1
13.d odd 4 1 2028.4.a.b 1
39.f even 4 1 468.4.a.a 1
52.f even 4 1 624.4.a.b 1
104.j odd 4 1 2496.4.a.d 1
104.m even 4 1 2496.4.a.m 1
156.l odd 4 1 1872.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.a.b 1 13.d odd 4 1
468.4.a.a 1 39.f even 4 1
624.4.a.b 1 52.f even 4 1
1872.4.a.i 1 156.l odd 4 1
2028.4.a.b 1 13.d odd 4 1
2028.4.b.d 2 1.a even 1 1 trivial
2028.4.b.d 2 13.b even 2 1 inner
2496.4.a.d 1 104.j odd 4 1
2496.4.a.m 1 104.m even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(2028, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 1024$$
$11$ $$T^{2} + 4624$$
$13$ $$T^{2}$$
$17$ $$(T - 14)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 72)^{2}$$
$29$ $$(T - 102)^{2}$$
$31$ $$T^{2} + 18496$$
$37$ $$T^{2} + 148996$$
$41$ $$T^{2} + 62500$$
$43$ $$(T - 140)^{2}$$
$47$ $$T^{2} + 87616$$
$53$ $$(T - 526)^{2}$$
$59$ $$T^{2} + 110224$$
$61$ $$(T + 410)^{2}$$
$67$ $$T^{2} + 355216$$
$71$ $$T^{2} + 774400$$
$73$ $$T^{2} + 256036$$
$79$ $$(T + 640)^{2}$$
$83$ $$T^{2} + 1904400$$
$89$ $$T^{2} + 2102500$$
$97$ $$T^{2} + 198916$$